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This question is based on attempting to construct the (homotopy type) of Lie groups using Cobordism Hypothesis style abstract nonsense.

There is an $\infty$-groupoid of smooth, framed manifolds where the isomorphisms are diffeomorphisms, the paths between isomorphisms are isotopies of diffeomorphisms, and so on. There is also a functor from this category of framed manifolds to the $\infty$-category of spaces sending each manifold $X$ to its fundamental $\infty$-groupoid $\Pi_\infty(X)$

Consider pairs consisting of a $\infty$-group $G$ (i.e. the loop space of some classifying space $BG$) acting on a framed manifold $X$ such that the action of $G$ on $\Pi_\infty(X)$ is homotopy equivalent to the canonical action of $G$ on $G$.

There is by construction such a pair for any compact Lie group: $G$ is the homotopy type of the Lie group and $X$ is the Lie group itself acted on the left by $G$ and equipped with the framing coming from acting by the Lie algebra on the right. Are there any other examples?

Charles Wang
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  • I haven't attempted a precise comparison with your question, but the theory of finite loop spaces and $p$-compact groups shows that quite weak homotopical data is sufficient to reconstruct a compact Lie group. – Neil Strickland Mar 31 '24 at 09:56
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    From @NeilStrickland suggestion, this answer is relevant as to when two Lie groups represent the same element in your construction. – Connor Malin Mar 31 '24 at 13:11
  • Ah this is very useful - this at least gives a resolution to my original motivation, which was to give a characterization of $\Pi_\infty (O(n))$ through pure abstract nonsense. – Charles Wang Mar 31 '24 at 18:56

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